Stochastic evidence aggregation system of failure modes utilizing a modified dempster-shafer theory

ABSTRACT

A system for obtaining diagnostic information, such as evidence about a mechanism, within an algorithmic framework, including filtering and aggregating the information through, for instance, a stochastic process. The output may be an overall belief value relative to a presence of an item such as, for example, a fault in the mechanism.

This application claims the benefit of a U.S. Provisional ApplicationNo. 60/888,997, filed Feb. 9, 2007.

U.S. Provisional Application No. 60/888,997, filed Feb. 9, 2007, ishereby incorporated by reference.

BACKGROUND

The present invention pertains to obtaining various pieces ofinformation, and particularly it pertains to the obtaining theinformation with an algorithmic framework. More particularly, theinvention pertains to aggregation of the various pieces of information.

SUMMARY

The invention is a system for obtaining diagnostic information, such asevidence, with an algorithmic framework, including filtering andaggregating the information through, for instance, a stochastic process.The output may be a belief value relative to time of a presence of anitem such as, for example, a fault in a mechanism.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 is a diagram of belief versus time to the present for variousfailure modes of a mechanism;

FIGS. 2 and 3 are portions of the diagram of FIG. 1 according to certainalgorithms, respectively;

FIG. 4 a is a basic block diagram of the present system;

FIG. 4 b is a diagram of an overview of the present approach;

FIG. 4 c is a diagram of a set of failure modes showing various areas ofcoverage by algorithms of the modes;

FIG. 5 is an operational block diagram of a model for the presentapproach;

FIG. 6 shows a table of elements of the model noted in FIG. 5;

FIG. 7 shows a table of scenarios from a predictive trend monitoringprogram used for testing against with evidence aggregation;

FIG. 8 is a graph of a number of failure modes versus a fault evolutiontime constants;

FIG. 9 has a table which lists sensors and corresponding noiseinformation for use in a simulator;

FIGS. 10-13 are graphs of diagnostic algorithm outputs for a particularscenario relevant to fault modes of a mechanism;

FIG. 14 shows time traces of a stochastic fusion output relative tocertain failure modes;

FIG. 15 shows a table of scenarios showing a comparison of theDempster-Shafer and stochastic fusion approaches;

FIG. 16 shows a table of the complexity metric for various scenarios;

FIG. 17 shows a table listing an accuracy metric for various scenariosand comparing the stochastic based fusion with the Dempster Shafer basedfusion;

FIG. 18 shows a table showing a variability metric for the stochasticand Dempster Shafer based fusions in view of various scenarios; and

FIG. 19 is a graph of the variability metric as a function of evidencedropout for stochastic and Dempster Shafer based fusions.

DESCRIPTION

Many decision support systems may rely on a series of algorithms thatprovide supporting evidence towards a specific conclusion. Architecturesthat combine outputs from multiple algorithms may be attempted under theguise of knowledge fusion or evidence aggregation. A basic premise ofsuch approaches is to combine conclusions from multiple algorithms toachieve desired levels of accuracy. Dempster's theory of aggregationprovides a practical solution using upper and lower probabilitybounds—minimizing the penalty of specifying all conditionalprobabilities used by a Bayesian belief network. Various heuristicextensions may be offered to handle the following limitations, albeitnot necessarily with success. First, some evidence aggregation theoriesdo not appear to handle causal relationships between conclusions, suchas those arising in cascading fault situations. Second, aggregationappears essentially static and thus previous evidence would be ignored.In many domains, the causal relationship between hypotheses appears verystrong. In fact, ignoring them may lead to incorrect conclusions. Thereappears a need to develop an approach for evidence aggregation that willhandle the noted limitations. To this end, an aggregation theory may bedeveloped based on stochastic processes and thus overcome thelimitations.

The aggregation issue may be viewed partially as a Markov process. Anoverall belief {right arrow over (x)}(t) may be modeled as a known, butunobservable state evolving in a linear state space. A probability ofcascading faults, if known, may be incorporated within a statetransition matrix. An incomplete universe may be modeled as an unknownstate with unknown dynamics. A diagnostic algorithm may provide a noisyobservation for the hidden states. Diagnostic aggregation may beequivalent to an estimation problem—where one may be estimating thehidden state {right arrow over (x)}(t) evolving in a partially knownstate space using multiple noisy observations Y_(A) _(j) (t). In asolution approach, one may handle these two elements through two coupledsteps. One may be filtering. In this step, one may handle the noiseassociated with evidence provided by various diagnostic algorithms.Since the diagnostic state space {right arrow over (x)} appears largerthan the algorithm expertise, only the relevant states would be updated.One may use a Kalman filter approach to handle the noise and update thenoise covariance matrices. Another may be aggregation. In this step, onemay handle the partial overlap associated with the evidence. In otherwords, one may aggregate the smooth updates from the various diagnosticalgorithms.

The present invention may be at least in part a mechanism foraggregating evidence provided by diagnostic algorithms. One may assumethat the output from a diagnostic algorithm is a stochastic realizationof the underlying failure mechanism. In other words, diagnosticalgorithms may provide a noisy and incomplete view (i.e., partiallyknown) of the diagnostic state. As an example, weather channels mayprovide partial observations of the true weather. The information may benoisy because the weather instruments have an inherent noise. This noiselevel may be reflected in the observation provided by the weatherchannel.

A machine could fail in 200 different ways but a diagnostic algorithmmay be able to provide observation for only one failure mode (FM) or 5failure modes. There may be noise because the pertinent sensors arenoisy.

The “what” here may be an approach for aggregating such information andderiving at a diagnostic state of the machine or system underconsideration. The “how” is the using a theory of filtering andaggregation for getting an estimate of this diagnostic state usingsuccessive observations from these diagnostic algorithms. The Bayesiantheory may be used for conflict resolution. Alternatively simpleaveraging scheme can also be used. One may have diagnostic aggregationor fusion which could be an overlap of two algorithms. If one algorithmhas a 70 percent level of confidence and the other has a 30 percentlevel of confidence of a broken blade, then there may be an overallweighted average that is an 80 percent level of confidence about thebroken blade; however, the level may still be less than 100 percent.

FIG. 1 is a graph that relates to a diagnostic algorithm of a jetengine. This algorithm may know of six failure modes out of a possible200 failure modes of the engine and hence provides partial observations.At the five unit point of time there may be a 0.1 belief (i.e., aconfidence level) in one of the failure modes (viz., failure mode 4(FM4)) though the belief may be less for the other failure modes. In thegraph of FIG. 2, there may be a zero belief in the failure mode 4, inthat belief in this mode does not appear in the graph. In FIG. 3, thehighest belief may be in the fourth failure mode diagnostics output atthe various points in time.

For instance, there may be two algorithms, A_(i), A_(j), reflectingdifferent beliefs according to FIGS. 2 and 3, respectively. The weightedaverage approach may provide a fusion or aggregation of the beliefsprovided by the algorithms to come up with one resultant belief. Theremay be obtained an overall or common belief value over time. The Kalmanfilter may be used to come up with a smooth value of the belief. Afeature of the present invention is coming up with a common belief ofthe two algorithms over a certain period of time.

FIG. 4 a is a block diagram of the present system. A diagnostic module91 may have an output connected to a conditioner module 93. An output ofmodule 93 may go to an aggregation module 94. Module 94 may provide anoutput 95 which may be a diagnostic state relative to time. Module 93may be a filter such as a Kalman filter. The aggregation may be that ofa weighted average type of fusion.

FIG. 4 b is a diagram of an overall approach of the present system. TheKalman filter 22 may take into account most or all of the availablehistory in a system 20 as illustrated in FIG. 4 b. The Kalman filterapproach may be used to handle the noise and update noise covariancematrices. The Kalman filter is an illustrative example in that anotherkind of filter may be used. A diagnostic module 34 may have an outputconnected to filter 22 and a prognostic module 37 may have an outputconnected to the filter 22. System 20 illustrates an example of thepresent approach. Module 37 may include a diagnostic algorithm module21. Module 34 may include a delay operator 24 which is Z−1 and ademultiplexer 28. Also, entering delay operator 24 may be an input{right arrow over (x)}(t) which includes diagnostic states 25 from anevidence aggregation module 26. An initial diagnostic state 23, which is{right arrow over (x)}(0), may go to the delay operator 24. This input23 is also called as the initial values for the diagnostic states.Module 26 may effectively provide an output 25 of system 20, which isthe diagnostic states for the entire system.

An output 27, {right arrow over (x)}(t−1), from delay operator module 24may go to the demultiplexer 28. {right arrow over (x)}(t−1) may indicatea delay of one time unit (e.g., a minute, hour, day, or the like).Demultiplexer 28 may relate to an algorithm A_(j), which need only see apart of a state. The algorithm A_(j) may see, for example, only fivefailure modes even though there may be 200 ways of breakage or failuremodes of the mechanism at issue, such as an engine. The demultiplexer 28may provide a selective operation by, for example, letting just the fivefailure modes and/or information about them through as an output 29,{right arrow over (x)}_(A) _(j) (t−1), to the Kalman filter 22. {rightarrow over (x)}_(A) _(j) (t−1) may be a value of the state from theimmediately previous calculation when t>1 or it may be X(0) when t=0. Itmay be the overall belief calculated from the previous step of evidenceaggregation 26. The demultiplexer 28 may provide a selective operationby selecting only the belief values for only the fibe failure modespertaining to the diagnostic algorithm A_(j).

Diagnostic algorithm A_(j) may do its own process. The algorithm couldbe a processing mechanism that comes up new observations for the beliefin these five failure modes, signal {right arrow over (y)}_(A) _(j) (t)respectively. This observation provides an input 31 to the KalmanFilter. As an example, the belief values for five diagnostic states fromthe previous calculation may be 0.1, 0.02, 0.0, 0.6, 0.01. This previousvalues provides input 29 to the Kalman Filter. At time t, the DiagnosticAlgorithm Aj may provide a new set of belief for only these five failuremodes based on its internal processing module 21. The Kalman filterreceives noisy observations 31 such as 0.2, 0, 0, 0.5, 0.3. The Kalmanfilter thus receives two inputs: input 28 which is the belief for thefive diagnostic states from a previous calculation, and input 31 whichis the belief provided by Algorithm Aj during the current calculationtime. These two inputs are processed by block 22 to generate a new setof values for the belief in only these five diagnostic states. Thisoutput 32 {right arrow over ({circumflex over (x)}_(A) _(j) (t) will bean updated belief value for only the five diagnostic states. An output32, {right arrow over ({circumflex over (x)}_(A) _(j) (t), may go fromthe Kalman filter 22 to the evidence aggregation module 26.

There may be more algorithms that do a similar operation but fordifferent failure modes, yet a union of the failure modes is within atotal 200 failure mode super set. This is illustrated in FIG. 4 b usinglayers 33. Each layer corresponds to one Diagnostic Algorithm, eachlayer consisting of module 21 that provides providing signal 31, module28 that selects relevant belief values from the previous calculationsand a Kalman Filter block 22. Each layer may provide a partial stateupdate, signal 32 corresponding to the failure modes that the diagnosticalgorithm Aj is aware of. The output provided by several layers 33 maybe partially overlapping but the union of all outputs may not exceed the200 failure modes for the system. There could be, for instance, fivealgorithms from five layers, respectively, providing individual beliefson one failure mode such as dirt in a pipe. But each algorithm couldhave beliefs on several failure modes but not necessarily the samefailure modes as the other four algorithms.

FIG. 4 c shows a diagram of the 200 failure modes 36, i.e., the superset, for the system being observed within a circle 35. The failure modes36 may be represented by “+”s in the circle 35. Each of the failuremodes 36 may be different from one another. Algorithms 1, 2, 3, 4 and 5may be represented by circles 41, 42, 43, 44 and 45, respectively, incircle 35. An overlap of two or more algorithm circles may cover one ormore failure modes 36 common to the respective algorithms. Thealgorithms here appear not to be exhaustive as to covering all of thefailure modes, as each of the algorithms covers just certain failuremodes. For illustrative purposes, the present discussion showsalgorithms covering less failure modes 36 than shown in FIG. 4 c. Thereare a number of {right arrow over ({circumflex over (x)}_(A) _(j) (t)values, one for each layer 33, for all of A_(j), ε{A_(j)}, where thelatter represents algorithm sets.

$\begin{matrix}{{\overset{\hat{arrow}}{x}}_{A_{1}}(t)} \\{{\overset{\hat{arrow}}{x}}_{A_{2}}(t)} \\{{\overset{\hat{arrow}}{x}}_{A_{3}}(t)} \\\vdots \\{{\overset{\hat{arrow}}{x}}_{A_{n}}(t)}\end{matrix}$The “n” number of noted items of evidence relative to the algorithms maybe aggregated. These items may be fused using the Dempster Shafer oranother approach. {right arrow over ({circumflex over (x)}_(A) _(i) and{right arrow over ({circumflex over (x)}_(A) _(j) may have severalcommon trajectories of the trajectories 11, 12, 13, 14 and 15 shown inFIGS. 2 and 3 of the respective failure modes which are similarlydesignated. There may be a range or an expertise covered by a particularalgorithm. The updates {right arrow over ({circumflex over (x)}_(A) ₁ ,{right arrow over ({circumflex over (x)}_(A) ₂ , {right arrow over({circumflex over (x)}_(A) ₃ , . . . , {right arrow over ({circumflexover (x)}_(A) _(n) , corresponding to the respective algorithms, may befused resulting in one vector {right arrow over (x)}(t).

For each different failure mode, one may get one belief number or value.Thus, there may be nine belief numbers from an algorithm for ninedifferent failure modes. One specific failure mode, dirt in a pipe,viz., a pipe failure, may be covered by five algorithms having fivebelief numbers which result in one belief number for that particularfailure mode. However, there may be more or less 120 failure modes 36 incircle 35 outside the algorithm coverage. These failure modes may beregarded as unknown and are not to be considered separately as they arecovered by an algorithm but may be considered collectively as one entityas represented by θ_(o).

The evidence aggregation module 26 in system 20 of FIG. 4 b may pull theinformation 32, {right arrow over ({circumflex over (x)}_(A)_(1 through n) (t), from filter 22 together for the “n” various layers33, to provide an estimated diagnostic state(s) 25, {right arrow over({circumflex over (x)}(t), of the system 20. The hat “^” of {circumflexover (x)}(t) may be dropped (resulting in x(t)), or it may be retained,when doing evidence aggregation. At the output of module 26, one may geta diagnostic state of the system at time (t) which goes back into theentire calculation at the next time step. Then time may change and theprocess may be repeated for each step of time. Thus, {right arrow over({circumflex over (x)}(t) or {right arrow over (x)}(t) may be plotted.

FIGS. 10-13 show the Y_(A) _(j) (t) (relative to engine cycles) of theinput 31 to filter 22 in FIG. 4 b. FIG. 14 shows plots after the fusionof {right arrow over ({circumflex over (x)}(t) (relative to simulationtime) for certain failure modes. FIGS. 10-13 reveal a raw input and FIG.14 shows the output after processing. As to the [39 41] trace 85 in FIG.14 where the failure modes are together, the relevant algorithm couldnot distinguish between these failure modes. There may be missingevidence and yet there is an output since the aggregation is robust.FIG. 19 shows a standard deviation in belief, ζ₂, of 0.05 where aboutone percent of the evidence is missing. A 10 to 15 percent dropout ofsensors may be normal for the state of the art. One may note in FIG. 19that the curve 47 representing the Dempster Shafer fusion standarddeviation in belief, ζ₂, rises significantly with an increase of freeevidence dropout, φ. The curve 48 representing the Stoch fusion standarddeviation in belief, ζ₂, rises significantly less than curve 47 with theincrease of free evidence dropout, φ. The lower curve 48 which shows asmaller deviation in belief which is more favorable since it is desiredthat performance of the system's evidence aggregation be stable.

The issue of evidence aggregation may arise in fault diagnosis usingmultiple classifiers and decision making using multiple experts. Withthis issue, one may be given a set of failure hypotheses H and a set ofalgorithms A. At each time t, each algorithm a_(j)εA provides evidencey_(a) _(j) (t), which assigns a measure of belief to elements in H. Thetheory of evidence aggregation provides a mechanism for calculating anoverall belief {right arrow over (x)}(t) for each element in H at timet. Presented here is a theory for evidence aggregation. In this theory,the aggregation problem is formulated as an estimation/filteringproblem. The aggregation problem is viewed as a partially known Markovprocess. Overall belief {right arrow over (x)}(t) is modeled as a known,but unobservable state evolving in a linear state space. Diagnosticalgorithm a_(j) provides noisy observation Y_(a) _(j) (t) for the hiddenstates {right arrow over (x)}(t). One may demonstrate the performance ofthe present approach with respect to its accuracy and variability underconditions of sensor noise and diagnostic algorithm drop-out. Further,one may provide empirical evidence of convergence and managing thecombinatorial complexity associated with handling multiple faulthypotheses.

As indicated herein, the aggregation issue may be viewed as a partiallyknown Markov process. Overall belief {right arrow over (x)}(t) may bemodeled as a known, but unobservable state evolving in a linear statespace. As an example, a system with six known failure modes may beconsidered. FIG. 1 indicates a belief in a presence of faults modeled asa stochastic process. The Figure illustrates the belief associated withvarious failure modes over time. Typically, a time axis may becommensurate with the physical system and the rate at which the faultmanifests. One may model the belief in the presence of a failure mode asa continuous variable rather than a discrete variable. Further, one maymodel this variable as a stochastic Markov process. A probability ofcascading faults, if known, may be incorporated within the statetransition matrix, x(0)=0. An incomplete universe may be modeled as anunknown state with unknown dynamics.

In FIG. 1, a belief relative to six failure modes 11, 12, 13, 14, 15 and16 (i.e., first through sixth failure modes) are shown over time from,for example, five hours through twenty-one hours, i.e., up to thepresent time, “Now”. Units of such a plot may be other than hours,depending on the starting date and kind of observations desired of theplot.

A diagnostic algorithm A_(j) may provide noisy observation e_(a) _(j)(t) for the hidden states {right arrow over (x)}(t). An algorithm A_(j)may provide a noisy observation for a smaller subset of all possiblediagnostic states. Similarly, algorithm A_(j) may provide a belief foran overlapping set of states. FIGS. 2 and 3 illustrate the noisyobservation provided by diagnostic algorithms A_(i), A_(j). Diagnosticaggregation may be equivalent to an estimation problem—where one isestimating the hidden state {right arrow over (x)}(t) evolving in apartially known state space using multiple noisy observations e_(a) _(j)(t). One may formulate the estimation as a Kalman filtering problem withstandard assumptions about state and observation noise.

FIGS. 2 and 3 indicate that diagnostic algorithms may provide noisy andpartial realization of the Stochastic process until a present time t.FIG. 2 shows belief provided by algorithm A_(i) relative to failuremodes 11, 23 and 15, relative to the time of five through twenty-onehours (i.e., the present moment). FIG. 3 shows belief provided byalgorithm A_(j) relative to failure modes 11, 12, 13 and 14, relative tothe same period of time of FIGS. 1 and 2.

A fusion mechanism 20 is shown in FIG. 4 b. The basic premise is thatdiagnostic algorithm A_(j) module 21 provides a noisy and partialrealization of the underlying stochastic process that describes thebelief with respect to various failure modes. In the present approach,these two elements may be handled through two coupled steps. One isfiltering. In this step, one may handle the noise associated withevidence provided by various diagnostic algorithms. Since the diagnosticstate space x is larger than the algorithm's expertise, just therelevant states are updated. Mathematically, relevant elements of statevector {right arrow over (x)}(t−1) at time t−1 may be updated using theoutput from A_(j) algorithm. That is, a partial {right arrow over({circumflex over (x)}_(A) _(j) (t) may be generated using {right arrowover (x)}(t−1) and an algorithm output Y_(A) _(j) (t). One may use aKalman filter may be used to handle the noise and update the noisecovariance matrices.

The second step may involve handling the partial overlap associated withthe various pieces of evidence. In other words, one may aggregate thesmooth updates {right arrow over ({circumflex over (x)}_(A) _(j) (t)provided by various diagnostic algorithms. Mathematically, in this step,one may calculate {right arrow over (x)}(t) using all of the available{right arrow over ({circumflex over (x)}_(A) _(j) (t).

A state space model may be formulated. One may be given a set of systemsexhaustive conditions Θ={θ₁, θ₂, . . . , θ_(n)}. As an example, θ₁, θ₂,. . . may represent various fault conditions in a large system. Theexhaustivity (closed world) assumption is not necessarily fundamentalbecause one may close any open world theoretically, by including into itan extra element θ₀ corresponding to all missing or unknown conditions.One may define a set S(Θ) such that:

1. θ₁, . . . , θ_(n)εS(Θ);

2. If s_(i), s_(j)εS(Θ), then s_(i)^s_(j) εS(Θ); and

3. No other elements belong to S(Θ), except those obtained by usingrules 1 or 2.

There may be N

2^(n)−1 elements in S(Θ). The state of the system may be described byassigning a belief to each element in S(Θ). One may let {right arrowover (x)}(t) denote the belief vector such that:

1. {right arrow over (x)}_(k)(t) is the belief assigned to the kthelement in S(Θ);

$\begin{matrix}{{{2.\mspace{14mu}{{\overset{arrow}{x}}_{k}(t)}} \geq 0},\;{{\forall k} = 1},\ldots\mspace{11mu},{N;{and}}} \\{{3.\mspace{14mu}{\sum\limits_{k - 1}^{N}{{\overset{arrow}{x}}_{k}(t)}}} = 1.}\end{matrix}$

{right arrow over (x)}(t) may be the state vector that describes thesystem at time t. Here, since the elements of S(Θ) correspond to failuremodes, {right arrow over (x)}(t) may describe the health state. A taskof knowledge fusion may be to calculate {right arrow over (x)}(t) usingmeasurements provided by various diagnostic and prognostic algorithms.

A basic premise of the present approach is to describe the health stateas a stochastic process. Specifically, one may model the health statesas a first order linear Markov chain. That is,{right arrow over (x)}(t)=F(t){right arrow over (x)}(t−l)+w(t); t>0,{right arrow over (x)}(0)={right arrow over (0)},  (1)where matrix F(t) corresponds to a dynamic evolution of the process andw(t) is the process noise. In the present model, F_(j,j)(t) mayrepresent that the system health at time t will remain in that health attime t+1. Off diagonal elements may capture the cascade or faultpropagation. For example, if F_(j,k)(t)≠0 may represent a cascadebetween the jth and the kth failure mode in the set S(Θ). Process noisemay be assumed to be normally distributed and stationary. That isw(t)˜N(0,Q).

An algorithm model may be formulated. M algorithms assumed, where A_(i)represents the ith algorithm. Each algorithm may have a predefined rangedenoted by r(A_(i)) which describes its expertise. Algorithm A_(i) maysatisfy the following conditions.

-   1. r(A_(i))⊂S(Θ), ∀_(i)=1, 2, . . . , M;-   2. I(r(A_(i))) is defined as the set of singletons within the ith    algorithm range;

${{3.\mspace{14mu}\bigcup_{i}^{M}{I( {r( A_{i} )} )}} = \{ {\theta_{1},\ldots\mspace{11mu},\theta_{n}} \}};$

-   4. Without loss of generality, one may assume that r(A_(i)) may be    assumed to be an ordered set with respect to S(Θ) (that is, if    |r(A_(i))=N_(i) then r(A_(i)) contains the first N_(i) elements of    S(Θ)—it follows from the first condition that N_(i)≦N);-   5. At time t, algorithm A_(i) may generate a diagnostic output Y_(A)    _(j) (t) or a basic belief assignment (bba) to elements in r(A_(i));    and-   6. A basic belief assignment from the ith algorithm may provide a    noisy and partial observation of the system health state, namely,    {right arrow over (x)}(t) (that is, Y_(A) _(i) (t)_(j) is the noisy    belief assigned to the jth element of r(A_(i))).

One may adopt a short hand notation y^(i)(t) to denote the diagnosticoutput from algorithm A_(i) at time t. Since algorithm A_(i) provides anoisy and partial observation for the system health vector, one may havean equation,y ^(i)(t)=H ^(i) {right arrow over (x)}(t)+v(t),  (2)where H^(i) is a diagonal matrix and v(t) is the observation noise.Stated differently, algorithm A_(i) may provide a partial and noisyrealization of the stochastic system health state vector. Since thealgorithms assign a basic belief to each element in r(A_(i)), and thesystem health vector is also expressed as belief, then diagonal elementsof H^(i) may be equal to one. That is,

$\begin{matrix}{{{{1.\mspace{14mu} H_{k,k}^{l}} = {{1\mspace{14mu}{for}\mspace{14mu} 1} \leq k \leq N_{i}}};{H_{k,k}^{l} = 0}},{k < N_{i} \leq N},{and}} \\{{2.\mspace{14mu}{v(t)}} \sim {{N( {0,R^{i}} )}.}}\end{matrix}$

For the non-cascade fault case, one may interpret F_(j,j)(t) as the rateat which the jth health state changes with time. Alternatively,F_(j,j)(t)=1 implies an absorbing state, while F_(j,j)(t)=0 implies anon-reachable state. The numerical value of F_(j,j)(t) determines howfast the knowledge fusion output will reach its final value. A smallernumber implies longer time, while a larger value will take shorter time.The numerical value may depend on how often the states are updated, orthe sampling time. In the present case, the diagnostic state is updatedwhenever the diagnostic algorithms execute and provide a noisyobservation. A commonly imposed Boeing Company requirement for onlinediagnosis is the 8 out of 10 rule. In this requirement, an algorithmshould hold its output for a minimum of eight execution cycles beforesending a “failed” BIT. One may adopt this requirement and use thefollowing value for F_(j,j)(t).F _(j,j)(t)=0.8, ∀_(j)=1, . . . , N  (3)While F_(j,j)(t) signifies the transition of remaining in the jth state,and since F_(j,k)(t)=0, the process noise term w(t) signifies thetransition from the jth state to an unknown state. One may interpretthis as a fault state that one are not aware of. Given a diagnosticalgorithm A_(i), one may be aware of I(r(A_(i))) distinct faults.However, there may exist other failure modes θ_(n+1), θ_(n+2), . . .that this algorithm is unaware of. Clearly, there is no point inenumerating all unaware states, since the pth unaware state isindistinguishable from the qth unaware state. One may interpret w(t) asthe transition to this unaware health state. The term fault coverage isused in the diagnosis literature to indicate a weighted fraction ofaware and unaware states.

-   1. P(θ_(k)) may be defined as the probability of fault θ_(k).    Typically P(θ_(k)) may be specified as a reciprocal called mean time    to failure.-   2. I(r(A_(i))) may be defined as the set of singletons within the    ith algorithm range.

${3.\mspace{14mu}\Theta} = {{\bigcup_{i}^{M}{I( {r( A_{i} )} )}} = \{ {\theta_{1},\ldots\mspace{11mu},\theta_{n}} \}}$

-   may be defined as the set of singletons.-   One may have

${Q_{k,k}^{i} = {1 - \frac{\sum\limits_{i \in {I{({r{(A_{i})}})}}}{P( \theta_{i} )}}{\sum\limits_{i \in \Theta}{P( \theta_{i} )}}}};{\forall{k.}}$

R^(i) is the noise covariance matrix for diagnostic algorithm A_(i). Thekth diagonal term of this matrix is a measure of the variabilityalgorithm A_(i) has with respect to detecting the kth health statecontained in r(A_(i)). Often this number may be derived from part of thetraining process regarded as a confusion matrix. The confusion matrixmay be a set of two numbers which include false positives and misses.One may make several simplifying assumptions as in the following.

1.  R_(j, k)^(i) = 0,

-   that is, variability in the belief assignment to the jth and kth    state is not correlated. To some extent, virtually all diagnostic    algorithms transform raw and noisy sensor data into a basic belief    assignment. This assumption implies that sensor noise is    uncorrelated and the transformation affects all sensors equally.

2.  R_(j, k)^(i) = 1 − α,

-   ∀k., which implies that the variability in the belief assignment to    the jth and kth states is equal. One may relax this assumption if    the confusion matrix is specified for each element in r(A_(i)),    rather than for the entire set.

An aggregation scheme is noted. First, the partial state update may bedealt with. At time t>0, the system health is described using {rightarrow over (x)}(t). Algorithm A_(j) may use a series of sensormeasurements and generate y^(i)(t), which is its belief for elementscontained in r(A_(i)) at the next time step. In the present simplifiedformulation, one may ignore the exact nature of these sensors. That is,it may suffice to say that once the vector y^(i)(t) is known, it doesnot necessarily matter how the algorithm arrived at this value. In thepresent formulation, y^(i)(t) may provide a noisy observation (equation(1) for the evolving system health state (equation (1)). To this end,one may adopt Kalman filtering approach to update the system healthstate. Since r(A_(i))⊂S(Θ), it follows algorithm A_(i) can update only asubset of the system health.

For algorithm A_(i) with range r(A_(i)) and |r(A_(i))|=N_(i), the stateof the Kalman filter may be represented by two variables: 1) {circumflexover (x)}(t−1)

{right arrow over (x)}(t−1) (1:Ni), which may be the estimate of thestate of time t−1; and 2) P(t−1), which may be the error covariancematrix (a measure of the estimated accuracy of the state estimate). P₀is a known constant.

At each step, algorithm A_(i) may update the state using the recursivefilter equations.

-   1. Predicted state: {circumflex over (x)}(t|t−1)=F(t) {right arrow    over (x)}(t−1).-   2. Predicted state covariance: P(t|t−1)=F(t)P(t−1)F(t)^(T)+Q(t).-   3. Innovation or measurement residual: r(t)=y(t)−H(t){circumflex    over (x)}(t|t−1).-   4. Innovation or residual covariance:    S(t)=H(t)P(t|t−1)H(t)^(T)+R(t).-   5. Kalman Gain: K(t)=P(t|t−1)H(t)^(T)S(t)⁻¹.-   6. Updated state estimate: x(t)=x(t|t−1)+K(t)r(t).-   7. Update estimated covariance: P(t)=(I−K(t)H(t))P(t|t−1).

The evidence aggregation may be dealt with. At each time step,algorithms A_(i), i=1, . . . , M, may provide a partial state update.One may let (x^(i)(t), P^(i)(t)) denote the state update provided by theith diagnostic algorithm. It may follow from the previous section that(x ^(i)(t),P ^(i)(t))=F({right arrow over (x)}(t−1),P ^(i)(t−1)).  (5)That is, the state update may be a function of the diagnostic statevector {right arrow over (x)}(t) and the state covariance matrixP^(i)(t−1). The diagnostic state at t can be calculated as a weightedaverage of individual algorithm updates; the weight is inverselyproportional to the covariance matrix. That is:

$\begin{matrix}{{{\overset{arrow}{x}}_{j}(t)} = {\sum\limits_{i - 1}^{M}{{x_{j}^{i}(t)}( {P_{j,j}^{i}(t)} )^{- 1}}}} & (6)\end{matrix}$The above equation will not necessarily take into account a conflictbetween conclusions of the algorithms. This may be especially importantwhen the algorithms have different ranges of expertise. One may need toconsider whether or not a conflict between conclusions of the algorithmsis due to multiple faults occurring simultaneously, but not beingaccounted for due to differences in algorithm ranges. Thus, one shouldtake into account a possibility that a set of faults {

j^(i)}, such that j^(I) ∉ {

r(A_(i))} could be occurring simultaneously. To do that, one may providethe following approach.One may denote sup (x_(i)(t)) as set of algorithms such that for eachalgorithm A εsup(x_(j))x_(j) ^(A)≠0. One may assign a non-zero belief toa state x_(j)=

x_(j) _(k) if ∩sup(x_(j) _(k) )≡∅. State x_(j) may be referred to aconflict state. The nonzero belief may be calculated as an average ofbeliefs assigned to states x_(j) _(k) .

To identify all possible conflict states, the following iterativeprocedure may be performed. First, for each pair of the non-zero beliefstates, all two-element candidate conflict states may be determined bytaking a union of two states. In each successive pass, the candidatek-element sets for a set of k states may be generated by joining(k−1)—element sets of those states, and then deleting those that containany (k−1) subsets that were not generated in a previous pass. Theprocess may be continued until no more sets can be generated. It shouldbe noted that in each step, one may use domain knowledge to eliminateany sets that are known to have zero belief.

A predictive trend monitoring (PTM) simulator may be noted. A simulatormay be built to demonstrate a fusion approach. This simulator may bebased on a PTM application. The simulator may include of three basicbuilding blocks, a model, a PTMD and a Stoch. The meta model maydescribe the functional dependencies within a typical jet engine. ThePTMD may capture historical incidents recorded by the PTM program. TheStoch may capture dynamic data needed for evaluating the Stochasticknowledge fusion algorithm. Each of these building blocks is describedherein.

A model may be noted. Basic elements of the model 50 are shown in ablock diagram of FIG. 5. The diagram may be a model operational one. Arelationship between the blocks or modules may be shown with an arrowand an associated description as shown by item 73. The terms inparentheses denote MATLAB™ variable names. A components (logical) module61 may be connected with an outgoing arrow 62 of objectives to an oGoals(oGoals) module 63. Module 61 may also be connected with an outgoingarrow 64 of failures to a fault modes (hypothesis) module 65. A features(signals) module 66 may have an outgoing arrow 67 of analysis to analgorithms (tests) module 68. A measurements (tests) module 69 may havean outgoing arrow 71 of sensors to module 68. An outgoing arrow 72 of afault-range may go from module 68 to module 65.

A summary of the elements and a count for each the elements of the model50 is listed in a table of FIG. 6. There may be 121 components of aturbine with 410 failure modes. A unit that can be removed and replacedis a component. Associated with the model may be 37 algorithms, 21measurements and 11 features. The model may be expressed as a MATLAB™function as shown in a code 1 in the appendix.

The model may be built using an offline tool eXpress™, made availablefrom a vendor, DSI International. This is a graphical tool that may leta user enter basic information about the physical system, which may bean engine in the present case. This model (meta data) may capture thefollowing information.

First, the components make up the physical system. For example,components in a jet engine may include a turbine, compressor, and soforth. Often, one may hierarchically decompose the system intocomponents and sub-components. In the PTMD project, one may stop thedecomposition until one reaches a part number listed in the relevantillustrated part catalogue. That is, a component may be that which canbe removed, ordered from the catalogue and replaced.

Second, oGoals may be noted. These are known as operational goals. AnoGoal may capture the intended functionality of the component. Forexample, oGoal of the starter motor is to provide initial torque.Although this information is not necessarily used by any of thediagnostic or reasoning algorithms currently deployed within the PTMDapplication, it may help systematic building of the meta data modelitself.

Third, fault modes may be noted. These modes may be referred to as thefaults associated with each component. Each component may have more thanone fault associated with it. In some sense, one may use the term “faultmodes” and “failure modes” interchangeably. Fault mode information maybe collected from published FMECA documents. In addition to listing thefault mode name, the model may also include the a priori likelihood ofthe fault.

Fourth, algorithms may be noted. They may be the diagnostic algorithmsthat are currently deployed within the PTMD application. The algorithmsmay use a variety of sensors, techniques and mathematical models. Themodel does not necessarily capture the details, except for the sensorused, the range of expertise and the confusion matrix generated duringan offline development step.

Fifth, measurements may be noted. They are the measurements used byvarious diagnostic algorithms currently deployed within the PTMapplication. In a loose sense, the terms “measurement” and “sensor” maybe interchangeably. However, measurements may include raw sensor data,derived quantities by the data acquisition system, and operatorobservations like smell and color. The model may contain a listing ofsuch measurements.

Sixth, features may be noted. They may be the features used by variousdiagnostic algorithms deployed within the PTMD application. Examples offeatures may include MAX, RATE, and so forth. A feature may be aspecific attribute of a sensor measurement that is being used to accessthe current situation. The feature might be or not be used in theapplication.

PTMD scenarios may be reviewed. Evidence aggregation was tested againstthe following scenarios collected from a PTM program. Descriptions ofthese scenarios are provided herein. A hot section degradation-trendbased scenario may correspond to the most commonly occurring problem foran engine, namely hot section degradation. In an engine, a hot sectionrefers to the power plant, i.e., the sub-assembly that generates power.Currently, one may have a series of algorithms that monitor the exhaustgas temperature (EGT) when the engine is delivering maximum power. ThePTMD suite may also include algorithms that monitor the temperature whenthe engine is idling. This scenario may be used to demonstrate the basicnotion of knowledge fusion, namely, an aggregation of supportingalgorithms to increase belief in the power section.

Another scenario is a hot section degradation—jump based one. Thisscenario is similar to the hot section degradation-trend based scenario,except that here one may be monitoring a shift of jump in the exhaustgas temperature rather than the trend. Jumps in the EGT value may becaused by abrupt failures, which rules out faults with the air intake.Faults associated with the air intake tend to be more gradual. One maycompare and contrast this scenario with the hot sectiondegradation-trend based scenario to understand how knowledge fusionweighs in different features of the same sensor.

A hot section degradation—fp suppression scenario is also similar to thehot section degradation-trend based scenario. As stated herein, hotsection degradation is analyzed by monitoring the EGT values duringengine idling, maximum power and startup. However, in the fp—suppressionscenario, algorithms monitoring EGT during startup may trigger falsely.None of the other algorithms appear to trigger. One may look for falsepositive suppression in this scenario. Knowledge fusion should beeffective in suppressing the false alarm based on non-trigger ofalgorithms that monitor the EGT during idling and maximum power.

A hot section alg cba—simple clustering scenario may be to be analyzedwith the hot section degradation-trend based scenario. PTMD may includea series of algorithms to monitor hot section degradation. In thisscenario, all of these algorithms may trigger and provide supportingevidence. In some sense, this scenario provides an ideal case for hotsection monitoring. One may look for the maximum belief under this idealsituation.

A hot section alg cba—complex clustering scenario may be analyzed withhot section alg cba—simple clustering scenario. Given an ideal conditionnoted in the hot section alg cba—simple clustering scenario, thisscenario may calculate a delta-increase due to the addition of one morealgorithm. This additional algorithm may use the same set of sensors,but uses a different technique which is clustering. An objective of thisscenario is to calculate the cost benefit of adding a complex algorithm.One may look for a delta change in the belief.

A multiple fault hypothesis—debris and cooler scenario tests the abilityof knowledge fusion to handle a multiple fault hypothesis. In thisscenario, two failure modes may occur. One mode may be related to inletdebris and the other mode may be related to oil cooler. Two algorithmswith relatively low uncertainty may assess the presence of inlet debrisand oil cooler. The oil cooler algorithm may provide small evidencetowards debris. One may look for the presence of two fault hypothesis inthis scenario.

A multiple fault hypothesis—gen and starter scenario may also test theability knowledge fusion to handle multiple fault hypothesis. In thisscenario, the generator may be failed in addition to a failed starter.Typically both these line replaceable units may be tested duringpower-up and can indicate failure at the same time. Two strongalgorithms may provide supporting evidence towards these faulthypotheses. One may look for the presence of two fault hypothesis inthis scenario.

A dependent evidence—scv scenario may test the ability of knowledgefusion to aggregate dependent and semi-dependent evidence. A suite ofthree algorithms may detect problems associated with the surge controlvalve. All of these algorithms may use the same sensor, but usedifferent features of the sensor. One may take a close look at thebeliefs to understand how the present KF handles dependent evidence.

A dependent evidence—starter scenario may also test the ability ofknowledge fusion to aggregate dependent and semi-dependent evidence. Asuite of three algorithms may detect problems associated with thestarter motor. Two of the algorithms may use univariate sensor tests,whereas the third algorithm may use a multivariate analysis. However,the third algorithm may use the same sensors and same features. Thisscenario may answer the question whether the third algorithm adds anyindependent evidence to knowledge fusion. The fault scenarios aresummarized in a table of FIG. 7. The model may be loaded a MATLAB™function as described in the appendix (code 2).

In reference to FIGS. 7 and 16-18, the scenarios (Scen) may be referredto numerically in that “1” is the hot section degradation—trend based,“2”, is the hot section degradation—jump based, “3” is the hot sectiondegradation—fp suppression, “4” is the hot section alg cba—simpleclustering, “5” is the hot section alg cba—complex clustering, “6” isthe multiple fault hypothesis—debris and cooler, “7” is the multiplefault hypothesis—gen and starter, “8” is the dependent evidences—scv,and “9” is the dependent services—starter.

Dynamic data may be needed to simulate the evolutionary behavior ofvarious scenarios. The dynamic behavior may in turn depend on how theunderlying faults are evolving. The 410 failure modes do not necessarilyevolve at the same rate. An evolution describes how the failure mode maymanifest from its inception to its full failed state. This should not beconfused with the onset of the failure itself or the mean time betweenfailures. For example, load compressor heat shield crack failure mode(index=83) may initiate during the n engine cycle. However, it may taken+τ engine cycles before it is declared as a full failure. The dynamicdata may capture this τ. One may contrast this with mean time betweenfailures, which refers to the total number of engine cycles between twosuccessive occurrences of full heat shield failure. A small τ mayindicate rapidly evolving failures, while large τ value may indicateslow evolving failures. In the present approach, a number of domainexperts within the involved engines division were interviewed andtypical r values for all the 410 fault modes were obtained. Rather thanrequesting precise numbers, the question was simplified to reflectlimitations imposed by the current data collection system. Within thePTMD, a measurement may be available at the end of an engine run.Consequently an algorithm A_(j) may produce one piece of diagnosticevidence for each engine run. Hence, τ was requested with respect toengine cycles. Fast faults may manifest within one cycle, whileprogressive faults may evolve over series of engine cycles. FIG. 8 is ahistogram that captures typical τ values (i.e., fault evolution timeconstants) for the 410 failure modes included in the model. The graphreveals a number of failure modes versus a typical time constant, enginecycles. A graph bar at a constant of about 1 shows an inception of afast failure. A graph bar at a constant of about 24 shows an inceptionof a slow failure.

Diagnostic conclusions made by various algorithms may depend on how fastthe underlying failure is evolving, the internal thresholds used byindividual algorithms, and the noise introduced by the sensors beingused by the algorithms. In the present approach, the algorithms may betreated as a black-box and thus the simulator would not necessarilyaccount for the second factor. The table in FIG. 9 lists sensors andtheir corresponding noise for use in a simulator. The table appears toshow the necessary data for dynamic information to be provided as anEXCEL™ file. This EXCEL™ file may be imported using the MATLAB™function. Code 3 of the appendix lists the MATLAB™ import function.

A simulation drive or simulator may be run for the scenarios describedfor a user specified time duration. As described herein, the time t=0may indicate an onset of a failure mode, and t=∞ may indicate acompletely evolved failure. At this point, the driver does notnecessarily simulate pre-failure or no-failure conditions. Although itmay be relatively simple to simulate pre-failure conditions, it wouldprovide little value in demonstrating the present aggregation scheme.Code 4 in the appendix lists the M-file that drives the scenarios. Thesimulation driver needs to invoke the initialize simulator function toload the simulator.

FIGS. 10, 11, 12 and 13 show output traces from various algorithms forthe hot section degradation—fp suppression scenario. As the failureevolves within this scenario, diagnostic algorithms may change theirbelief in various fault hypotheses. Each of the traces 75, 76, 77 and 78corresponds to a specific fault hypothesis which is not shown in theFigures. In the Figures, an X-axis represents engine cycles while theY-axis represents {right arrow over (y)}_(A) _(j) (t). The individualtrend lines 75, 76, 77 and 78 represent the beliefs in the various faulthypotheses. A fault hypothesis index is not shown for illustrativepurposes, but the each of the curves 75, 76, 77 and 78 corresponds tothe same fault mode, respectively, in each subplot of the Figures. Forinstance, algorithms 52 and 51 of FIGS. 11 and 10, respectively, agreeon the fault mode depicted by curve 78, whereas algorithm 54 of FIG. 13appears to provide strong evidence towards the fault mode depicted bythe trend line 76, and algorithm 53 of FIG. 12 appears to provide strongevidence towards the fault mode depicted by the trend line 75. Thereappears to be no single dominant winner and thus a fusion scheme seemsnecessary to combine this evidence to a point where decisions may bemade.

Simulation studies may be broadly classified into two categories, eachof which is intended to evaluate the effectiveness of the presentstochastic fusion approach (abbreviated as “Stoch”) In the threestudies, the classical Dempster-Shafer (abbreviated as “DS”) fusion maybe used as a baseline. Knowledge fusion may be an important step fordownstream decision making—either for doing some isolation procedures orrepair actions. A key metric is accuracy. If the knowledge fusionconcludes that fault “A” is present, then this metric may measure theprobability that the system has fault “A”. The closer this number getsto 1, the more accurate is the conclusion. The knowledge fusion shouldmake this conclusion by aggregating all of the evidence provided byvarious diagnostic algorithms. Accuracy may be calculated by performinga series of Monte Carlo (MC) experiments or trials described herein.

Calculation of accuracy may depend on the overall setting of anexperiment. That is, I which is an index of the fault being simulated, Twhich is the simulation time duration [50], and M which is the number ofMonte Carlo trials [200]. For each experiment, both the DS-based fusionand the stochastic fusion may generate an output matrix {right arrowover (x)}(t), wherein the rows of this matrix may contain a belief invarious failure modes, and the columns contained the belief as thesimulation time t elapses. The next step may involve sorting the outputmatrix for each time t and retaining just the top 3 indexes. Then, onemay count the number the fault index being simulated is present. Anindicator E_(p) may be generated for each of the Monte Carlo trials. Asummary of the procedure is in the following items.

-   p'th trial Output: {{right arrow over (x)}(t)}_(p) 0≦t≦T, p≦1≦M-   Winner Index: {{right arrow over (s)}I(t)}_(p)=sort({{right arrow    over (x)}(t)}_(p))-   select(top 3 index)-   Indicator: E_(p)=count(find({{right arrow over (s)}I(t)}=I)),    11≦t≦T  (7)-   One may then define the accuracy metric for the Ith fault scenario    with the following.

$\begin{matrix} ϛ_{1}\Leftrightarrow{\frac{\sum\limits_{p = 1}^{M}E_{p}}{M \times ( {T - 11} )} \times 100}  & (8)\end{matrix}$

Given a baseline scenario for the Ith fault, a second metric thatmeasures the variability in the fusion output may be defined with thefollowing.

-   Baseline: {right arrow over (x)}0(t,I), 0≦t≦T-   p'th trial Output: {{right arrow over (x)}(t,I)}_(p), 0≦t<T, p≦1≦M-   p'th trial Dev: dv_(p)(t)={{right arrow over (x)}(t,I)}_(p)−{right    arrow over (x)}0(t,I)-   p'th trial Std: S_(p)=std(dv_(p)(t))-   Then one may define the variability metric for the Ith fault    scenario with the following.

$\begin{matrix} ϛ_{2}\Leftrightarrow\frac{\sum\limits_{p = 1}^{M}S_{p}}{M}  & (9)\end{matrix}$

As noted earlier, if a system has N fault modes, under a worst casescenario of all N faults occurring simultaneously, then thedimensionality of {right arrow over (x)}(t)=2^(N)−1. This number maygrow rapidly and beyond modest values of N, the computational complexitymay become intractable. However, if one imposes a maximum value M fornumber of simultaneous faults, the maximum dimensionality of the statespace may be

${\sum\limits_{l = 1}^{M}\begin{pmatrix}N \\i\end{pmatrix}},$where (_(i) ^(N)) denotes the binomial term “N choose i”. In simplerterms, if one allows a maximum of M simultaneous faults, then one shouldup all of the combinations of one simultaneous fault, two simultaneousfaults, all the way till M is reached. For the Ith fault scenario,|{right arrow over (x)}| may denote the number of such combinationsconsidered by Stoch-fusion. A complexity metric may be defined with thefollowing.

$\begin{matrix}{ϛ_{3} = \frac{\overset{arrow}{x}}{\sum\limits_{i = 1}^{M}\begin{pmatrix}N \\i\end{pmatrix}}} & (10)\end{matrix}$

As ζ₃ gets closer to 1, Stoch-fusion considers all possiblecombinations, making the in-built conflict resolution ineffective.Smaller values of ζ₃ may denote effective conflict resolution, and thusStoch-dea is reasoning with only those fault combinations that arepertinent. ζ₁ may tell one if the Stoch-dea is indeed retaining theright only. In short, ζ₃ may be used to measure the effectiveness of theconflict resolution discussed relative to the aggregation scheme herein.

A series of Monte Carlo experiments may be performed. An experiment maybe constructed by randomly varying the various factors including sensornoise, missing evidence and the PTMD scenario. As to sensor noise,random noise following a Gaussian distribution may be introduced in eachsensor. The mean of this noise may be assumed to be zero and thestandard deviation as listed in the table in FIG. 9. A MATLAB randnfunction may be used to introduce sensor noise. As a result of thenoise, diagnostic evidence produced by various algorithms may vary. Forthe purposes of this approach, the diagnostic algorithm A_(j) may beconsidered as a blackbox. Corresponding to sensor noise, at the presentend, one may say noisy values for {right arrow over (y)}_(A) _(j) (t) asdepicted in FIGS. 10-13.

As to missing evidence, an objective of this approach may be to test theability of the fusion approach to handle missing evidence. Missingevidence may be introduced as a random variable following a uniformdistribution. Evidence {right arrow over (y)}_(A) _(j) (t) produced byalgorithm A_(j) may have a φ chance of being absent. In other words,algorithm A_(j) has a φ chance of failing during each execution cycle.φ=0 indicates no failures or zero dropouts, MATLAB's rand function maybe used to generate a sequence of 0/1 for a specified value of φ foreach diagnostic algorithm A_(j). “1” may indicate successful executionand thus it may produce evidence. “0” may indicate failed execution andthus no evidence at time t. In an ideal sense, the knowledge fusionshould be robust relative occasional evidence dropouts caused byalgorithm execution failures.

As to the PTMD scenario, each of the scenarios discussed herein maysimulate different fault conditions wherein different diagnosticalgorithms may provide conclusions. It should be noted that thesescenarios are representative of typical behaviors seen from an onlinePTMD application. They may vary in complexity and thus hence it may notbe reasonable to compare the hot section degradation—trend basedscenario with the hot section degradation—jump based scenario. However,it may be believed that these scenarios collectively describe typicalbehavior and thus can be used for making generic claims about DS-basedfusion and the stochastic fusion.

For a given scenario, and a φ setting, the simulator may be run tosimulate the evolution of the underlying fault. Within a time window,each of the diagnostic algorithms may use its respective sensors andcalculate the belief in the presence of one of more faults. Thealgorithm A_(j) should generate a belief for only faults that it knowsabout. Variability in {right arrow over (y)}_(A) _(j) (t) may beintroduced because of sensor noise. t=0 at the start of the simulationrun and stochastic fusion may generate {right arrow over (x)}(t) usingthe aggregation scheme as described herein. The dimensionality of {rightarrow over (x)}(t) may change, depending on the fault scenario. FIG. 14shows the time trace of a stochastic fusion output, namely {right arrowover (x)}(t). Such trace may correspond to the hot sectiondegradation—jump based scenario and zero dropout. Trend lines maycorrespond to a belief in various fault modes. As indicated in thelegend of FIG. 14, the [36] trace 88 corresponds to the belief in thefault mode with an index 36 (viz., fault 36). The [41] trace 87corresponds to the belief in the fault mode with an index 41. The [3941] trace 85 denotes the belief in fault 39 and 41 being present in thesystem. Similarly, the [36 39] trace 86 denotes the belief in fault 36and 39 being present in the system. Stated differently, [A] may denotethat the corresponding fault is present by itself, whereas [A B] maydenote the belief that both fault A and B are present simultaneously.For example, in the PTMD multiple fault hypothesis—gen and starterscenario, the generator is faulty and the starter motor is also faulty.

Before running a series of Monte Carlo experiments, a simpler case maybe run to test multiple hypotheses testing. This approach may bedesigned to test the ability of stochastic fusion to handle multiplefaults, rather than noisy evidences. In other words, here no sensornoise is introduced and the condition may be simulated corresponding toa fully evolved fault. This case may be derived from the PTMD multiplefault hypothesis—debris and cooler scenario and multiple faulthypothesis—gen and starter scenario. The number of faults may be reducedto eleven, and six diagnostic algorithms may be created. Three of thesealgorithms may provide very strong evidence towards three distinctfaults with indexes 4, 7, and 11. Four scenarios may be simulated asdescribed in the table of FIG. 15. The table shows a typical output froma Monte Carlo trial. As shown in the table, when only fault 4 ispresent, both the Dempster-Shafer and stochastic fusion approaches mayindicate a clear winner, namely [4]. However, when fault 4 and 7 aresimulated simultaneously, a superior quality of the present stochasticfusion approach appears evident. Note that in this scenario, one mayexpect the stochastic fusion to conclude that both faults 4 and 7 arepresent. That is [4 7] and not merely [4] and [7] which indicate thatfault 4 or fault 7 may be present. The stochastic fusion approachconsiders the case of multiple faults being present. Although belief in[4 7] is not the maximum, it nevertheless is a top 3rd candidate. Theutility of the stochastic fusion approach appears clear. Since one mayhave 11 fault modes, there is a 2¹¹−1 possible combinations. Out ofthese combinations, Stoch fusion not only may pick the correctcombination, but also the belief in this combination may bubble up tothe top. DS-fusion on the other hand does not appear to make thisconclusion. As stated herein, 2 algorithms provide very strong evidencetowards fault 4 and fault 7 individually. The DS approach appears unableto resolve this direct conflict, and among the 2 strong algorithms, theslightly stronger algorithm has an edge. This is evident by the beliefassigned to [4] and [7]. This example highlights a present ability toresolve conflict as well as manage the combinatorial complexity. Thenext scenario illustrates the same theme, when fault 4 is present alongwith fault 11. Similar results may hold true for 3 simultaneous faults4,7,11 as shown in the table of FIG. 15.

It appears evident that Stoch-based fusion is able to resolve conflictsuccessfully and handle multiple faults. A question is how efficient wasthis conflict resolution may be. This is where one may look at the ζ₃metric. A table in FIG. 16 lists the complexity metric for variousscenarios. As noted herein, smaller values of ζ₃ are desired. Asindicated in the FIG. 16, the maximum number of simultaneous faulthypotheses is fixed to 3 to illustrate the computational attractivenessof the Stoch-based fusion. The conflict resolution has ability to limitthe number of combinations, and as shown herein, these combinationsappear accurate. On an average, just 16 percent of all possiblecombinations were considered, providing a ten-fold decrease in thecomputational complexity. One may conclude that Stoch-based fusion isable to handle multiple faults as well as manage the combinatoriallyexploding state space using conflict resolution. Another question is howthe Stoch-based fusion performs with respect to its accuracy. Here iswhere one may look at the ζ₁ metric. A table in FIG. 17 lists theaccuracy metric for various scenarios and compares the Stoch-basedfusion with the classical DS-based fusion. As noted herein, it isdesired that ζ₁→1. As one runs several MC trials, it may be noted thatthe average accuracy for both Stoch and DS based fusion deviate from 1.However, ζ₁ deteriorates rapidly for DS-based fusion as one begins toloose evidence. That is, when φ=0.2, DS-based fusion appears about 62percent accurate. In other words, in only 87 out of 100 times, DS-basedfusion appears able to correctly identify the underlying fault. Also,DS-based fusion failed for the scenarios of the multiple faulthypothesis—debris and cooler scenario and the multiple faulthypotheses—gen and starter, which contained multiple faults. If oneignores these two scenarios and recalculates ζ₁ for DS-fusion, then theaccuracy increases to 79.73, which is still smaller than thecorresponding number for Stoch-based fusion.

A casual look at FIG. 17 may indicate that Stoch-based fusion may haveless accuracy than DS-based fusion. Although the numbers appear toindicate such, this conclusion may be misleading. The aggregations usedby DS and Stoch are different and thus a comparison of dissimilarapproaches in this sense does not appear appropriate. An important pointis the ability of Stoch-based fusion to handle multiple faults as wellas have more graceful of performance as one begins to lose moreevidence. It is expected that the performance of any fusion method maydecrease as the quality of the evidence degrades. Stoch-based fusionappears to bring a more graceful degradation and thus permits one toprovide tighter bounds on real-life situations. A significance of thefiltering component of Stoch-dea is highlighted by the results shown inFIG. 17.

As noted herein, it appears evident that accuracy of Stoch-based fusiondegrades slowly compared to DS-based fusion. A question, in the caseswhere Stoch-based fusion appears accurate, is how much the beliefvaries. For an answer, one may look at the ζ₂ metric. Ideally, one wouldlike ζ₂→0. However, this will not happen since sensors contain noise andthe quality of the evidence deteriorates. A table in FIG. 18 providesempirical evidence for the existence of δ in a Stoch-based fusion. Itappears that one cannot establish a limit using theoretic derivations.FIG. 18 shows that the ζ₂ values for Stoch-based fusion appearconsistently smaller than those for DS-based fusion. By definition,since ζ₂ is calculated only when the algorithm is accurate, it appearnot applicable (n/a) for DS-based fusion relative to scenarioscontaining multiple faults.

ζ₂ appears to increase as the quality of evidence deteriorates (as maybe simulated by increasing values of φ). One may identify an upper boundδ such thatζ₂≦δ, ∀φ≦φ^(max), ∀ scenarios.  (11)Based on experimental studies, one could establish an empirical bound.FIG. 19 shows the values of ζ₂ as a function of φ. From an appearance ofthe trend line, existence of an upper bound for ζ₂ seems plausible.However, it appears that one can not derive such a limit based ontheoretical considerations.

Multiple hypotheses management is an important factor in automatic ormanual decision making. Such management may range from a military fieldanalyst trying to explain a series of pieces of surveillance evidence,or a business analyst trying to explain potential malfunction using aseries of pieces of financial evidence. The present fusion approach maybe extended beyond the diagnostic decision making examples discussedherein. A decision making process that involves multiple evidences whichevolve over time is a suitable candidate for the present fusionapproach. Such evidence may come from online or offline diagnosticalgorithms monitoring a jet engine, a complex refinery process, or otheritems. Simulation studies may show the ability of the present approachto handle noisy as well as intermittent evidence using a filteringscheme, manage combinatorial complexity by pruning the hypotheses statespace using the concept of conflict resolution, and aggregateoverlapping evidence using an averaging scheme.

In the present specification, some of the matter may be of ahypothetical or prophetic nature although stated in another manner ortense.

Although the invention has been described with respect to at least oneillustrative example, many variations and modifications will becomeapparent to those skilled in the art upon reading the presentspecification. It is therefore the intention that the appended claims beinterpreted as broadly as possible in view of the prior art to includeall such variations and modifications

1. An information fusion system comprising; at least two diagnosticalgorithms for providing belief values about a failure mode of amechanism; a filter for receiving the belief values from the diagnosticalgorithms and providing smoothed belief values; and an aggregator foraggregating the smoothed belief values into at least one diagnosticstate; and wherein: one diagnostic algorithm is coupled to a diagnosticmodule comprising a demultiplexer having an output connected to an inputof the filter; another diagnostic algorithm is coupled to an aggregationmodule having an input connected to an output of the filter; the filteris a Kalman filter; and the filter is for smoothing noise in the beliefvalues provided by the diagnostic algorithms; the filter uses adiagnostic state at time t−1 to smooth out noise in the belief valueprovided by a diagnostic algorithm at time t; the aggregating thesmoothed belief values from the filter takes into account a conflictamong evidence provided by the diagnostic algorithms; and multiplefailure mode hypotheses are generated by taking a pair-wise union ofelements in the conflict and iteratively taking pair-wise unions ofmultiple failure hypotheses generated during a previous iteration. 2.The system of claim 1, wherein the diagnostic state indicates an overallbelief value about the failure mode.
 3. The system of claim 1, whereinthe belief values are provided for particular instances of past and/orpresent time by the diagnostic algorithms.
 4. The system of claim 1,wherein the filter is for filling a gap in belief values resulting fromoccasional drop-outs in the belief values provided by the diagnosticalgorithms.
 5. The system of claim 1, wherein the filter is foraccounting for a propagation of faults within the mechanism.
 6. Thesystem of claim 1, wherein the filter uses a diagnostic state at timet−1 resulting from the aggregating.
 7. The system of claim 1, whereinthe conflict among the evidence is resolved by generating multiplefailure mode hypotheses.
 8. The system of claim 7, wherein the multiplefailure mode hypotheses cover two or more failures occurringsimultaneously.
 9. The system of claim 7, wherein some multiple failuremode hypotheses are restricted relative to prior knowledge about failuremode probabilities.
 10. The system of claim 1, wherein a diagnosticstate for single and/or multiple failure mode hypotheses set iscalculated by a weighted averaging of smoothed belief values.
 11. Thesystem of claim 1, wherein the aggregating is of data from a stochasticprocess.
 12. An evidence aggregation system comprising: a diagnosticmodule; a filter connected to an output of the diagnostic module; and anaggregation module connected to an output of the filter; and wherein:the diagnostic module comprises: a delay operator; and a demultiplexerconnected to the delay operator and the filter; the delay operator hasan input for receiving an initial diagnostic state; the delay operatorhas an output for providing a time delayed diagnostic state to thedemultiplexer; the demultiplexer has an output for providing selecteddiagnostic state information to the filter; the filter is for providingbelief values to the aggregation module; the aggregation module is forproviding diagnostic states; and a diagnostic state is related to afailure mode of an observed mechanism.
 13. A method for fusinginformation comprising: obtaining belief values via a diagnostic modulecomprising a demultiplexer about one or more failure modes of astochastic system; filtering the belief values into smoothed beliefvalues; and aggregating the smoothed belief values into at least onediagnostic state about the system; and wherein the filtering for fillinga gap in belief values resulting from occasional drop-outs in the beliefvalues provided by the diagnostic algorithms; the filter is foraccounting for a propagation of faults within the mechanism; and adiagnostic state for single and/or multiple failure mode hypotheses setis calculated by a weighted averaging of smoothed belief values.